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Veryfing ODE for complicated y(t)

  1. Jan 28, 2012 #1
    1. The problem statement, all variables and given/known data
    For the differential equation, verify (by differentiation and substitution) that the given function y(t) is a solution.

    2. Relevant equations
    [itex] y' - 4ty = 1 [/itex]

    [itex] y(t) = \int_{0}^{t} e^{-2(s^{2}-t^{2})} ds[/itex]

    3. The attempt at a solution
    I attempted to take [itex]\frac{d}{dt}[/itex] of y(t) as usual but
    1. if I do not try bringing the [itex]\frac{d}{dt}[/itex] inside the integral I can do nothing because there is no elementary antiderivative of y(t).
    2. if I do bring the [itex]\frac{d}{dt}[/itex] inside the integral, I can use the chain rule to get
    [itex] y(t) = (-2) \int_{0}^{t} (-2t) e^{-2(s^{2}-t^{2})} ds[/itex]
    but since my variable of integration is ds not dt, this doesn't allow me to use a u-substitution as I had hoped nor can I think of a way to relate ds and dt.

    More or less I do not know how to take [itex]\frac{d}{dt}[/itex] of y(t) and I do not know any other ways to solve the problem.
    Last edited: Jan 28, 2012
  2. jcsd
  3. Jan 28, 2012 #2
    Do you have to do it that way? Because it looks like you can solve it linearly.
  4. Jan 28, 2012 #3
    Yes, the problem asks to verify that y(t) is a solution to the differential equation y' + 4ty = 1.
  5. Jan 28, 2012 #4


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    I don't think you have to be able to do the integral to identify that expression as [itex]4 t y(t)[/itex]. Since you are integrating ds you can factor out the t.
  6. Jan 28, 2012 #5

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